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  • src/documentation/constructs/tesselation.dox

    r8544b32 rcaece4  
    2020 * to such a surface.
    2121 *
    22  * \section tesselation-procedure Tesselation procedure
     22 * \section tesselation-procedure
    2323 *
    2424 * In the tesselation all atoms act as possible hindrance to a rolling sphere
     
    3434 * other (although the code for that is not yet fully implemented).
    3535 *
    36  * \section tesselation-convexization Making a surface convex
    37  *
    38  * A closed surface created by the aforementioned procedure can be made convex
    39  * by a combination of the following two ways:
    40  * -# Removing a point from the surface that is connected to other points only
    41  *    by concave lines. This might also be imagined as removing bumps or
    42  *    craters in the surface.
    43  * -# flipping a base line or rather adding a general tetrahedron to remove a
    44  *    concave line on the surface.
    45  *
    46  * With the first way one has to pay attention to possible degenerated lines
    47  * and triangles. That's why prior to the this convexization procedure all
    48  * possible degenerated triangles are removed. Furthermore, when looking at
    49  * a removal candidate and its connected points, all these points are split
    50  * up into so-called connected paths. The crater to be removed or filled-up
    51  * has a low point -- the point to be removed -- and a rim, defined by all
    52  * points connected by concave lines to the low point. However, when a point
    53  * has degenerated lines attached to it (i.e. two lines with the same
    54  * endpoints), it may have multiple rims (imagine two craters on either
    55  * side of the surface and the volume being so small/slim that they reach
    56  * through to the same low point). We have to discern between these multiple
    57  * rims, therefore the connected points are placed into different closed
    58  * rings, so-called polygons. The point is the removed and the polygon re-
    59  * tesselated which essentially fills the crater.
    60  *
    61  * With the second way, we have to pay attention that the filled-in tetrahedron
    62  * does not intersect already present triangles. The baseline defines two
    63  * points and as the tesselated surface is closed, it must be connected to two
    64  * triangles. These together define a set of four points that make up the
    65  * tetrahedron. Naturally, two sides of the tetrahedron are always present
    66  * already (and will become removed in place of the other two, effectively
    67  * adding more volume to the tesselation).
    68  * Now first, we only flip base lines that are concave. Second, none of the two
    69  * other sides of the tetrahedron must be present. And lastly, we must check
    70  * for all surrounding triangles that the new baseline (formed by point 3
    71  * and 4) does not intersect these. Essentially, if we imagine a plane
    72  * containing this new baseline, then each possibly intersecting triangle shows
    73  * up as a brief line segment. We have to make sure that all of these segments
    74  * remain below the new baseline in this plane. Also, things are complicated
    75  * as the first and last line segment will always intersect with the baseline
    76  * at the endpoint. There, we basically have to make sure that the line segment
    77  * goes off in the right direction, namely outward.
    78  *
    79  * \section tesselation-volume Measuring the volume contained
    80  *
    81  * There is no straight-forward way to measure the volume contained in a
    82  * non-convex tesselated surface. However, there is for a convex surface
    83  * because convexity means that a line between any inner point and a point on
    84  * the boundary will not intersect the surface anywhere else. Hence, we may
    85  * use the center of gravity of all boundary points (by the same argument it
    86  * must be an inner point) and calculate the volume of the general tetrahedron
    87  * by looking at each of the tesselation's triangles in turn and summing up.
    88  *
    89  * We can calculate the volume of the original non-convex tesselation because
    90  * the two ways mentioned above -- removing points and flipping baselines --
    91  * both involve just addding general tetrahedron whose volume we may easily
    92  * calculate. By bookkeeping of how much volume is added and calculating the
    93  * total convex volume in the end, we also get the volume contained in the
    94  * prior non-convex surface.
    95  *
    96  * \section tesselation-extension Issues whebn extended a tesselated surface
     36 * \section tesselation-extension
    9737 *
    9838 * The main problem for extending the mesh to match with the normal sense is
     
    10747 * from a combination of spheres with van der Waals radii.
    10848 *
    109  * \date 2014-10-09
     49 * \date 2014-03-10
    11050 *
    11151 */
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